A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

1, 3, 2, 4, 5, 6, 10, 9, 8, 7, 11, 12, 13, 14, 15, 21, 20, 19, 18, 17, 16, 22, 23, 24, 25, 26, 27, 28, 36, 35, 34, 33, 32, 31, 30, 29, 37, 38, 39, 40, 41, 42, 43, 44, 45, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset: 1

Author

Clark Kimberling, Aug 01 2000

Keywords

• nonn
• tabl
• nice
• easy

Comments

A triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are increasing; and (4) even-numbered rows are decreasing.

Self-inverse permutation of the natural numbers.

Mirror image of triangle in A056023. - Philippe Deléham, Apr 04 2009

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Dec 24 2012

For generalizations see A218890, A213927. - Boris Putievskiy, Mar 10 2013

Examples

			The start of the sequence as a table:
   1,  3,  4, 10, 11, 21, ...
   2,  5,  9, 12, 20, 23, ...
   6,  8, 13, 19, 24, 34, ...
   7, 14, 18, 25, 33, 40, ...
  15, 17, 26, 32, 41, 51, ...
  ...
Enumeration by boustrophedonic ("ox-plowing") diagonal method. - _Boris Putievskiy_, Dec 24 2012
The start of the sequence as triangle array read by rows:
   1;
   3,  2;
   4,  5,  6;
  10,  9,  8,  7;
  11, 12, 13, 14, 15;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of table, flattened
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
• Eric W. Weisstein, MathWorld: Pairing functions
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A079826, A131179 (first column), A218890, A213927.

Cf. A000027, A038722.

Programs

Haskell

a056011 n = a056011_tabl !! (n-1)
a056011_list = concat a056011_tabl
a056011_tabl = ox False a000027_tabl where
  ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss
a056011_row n = a056011_tabl !! (n-1)
-- Reinhard Zumkeller, Nov 08 2013

Maple

A056011 := proc(n,k)
        if type(n,'even') then
                A131179(n)-k+1 ;
        else
                A131179(n)+k-1 ;
        end if;
end proc: # R. J. Mathar, Sep 05 2012

Mathematica

Flatten[If[EvenQ[Length[#]],Reverse[#],#]&/@Table[c=(n(n+1))/2;Range[ c-n+1,c],{n,20}]] (* Harvey P. Dale, Mar 25 2012 *)
With[{nn=20},{#[[1]],Reverse[#[[2]]]}&/@Partition[ TakeList[ Range[ (nn(nn+1))/2],Range[nn]],2]//Flatten] (* Harvey P. Dale, Oct 05 2021 *)

Formula

a(n) = ((i+j-1)*(i+j-2)+((-1)^t+1)*i - ((-1)^t-1)*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

Extensions

New name from Peter Luschny, Apr 15 2023, based on Boris Putievskiy's comment